IJMMS 2004:32, 1703–1714
PII. S0161171204007392
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ON THE EXPANSION THEOREM DESCRIBED
BY H(A, B) SPACES
A. A. EL-SABBAGH
Received 28 February 2000
The aim of this paper is to construct a generalized Fourier analysis for certain Hermitian
operators. When A, B are entire functions, then H(A, B) will be the associated reproducing
kernel Hilbert spaces of Cn×n -valued functions. In that case, we will construct the expansion
theorem for H(A, B) in a comprehensive manner. The spectral functions for the reproducing
kernel Hilbert spaces will also be constructed.
2000 Mathematics Subject Classification: 47L25.
1. Introduction. Let H be the Hilbert space over C with inner product [·, ·] defined
b
by [f , g] = a g(t)f (t)dt, f , g ∈ H, where denotes the complex conjugate. Let H 2
be the Hilbert space of all ordered pairs {f , g}, where f , g ∈ H with the inner product
·, · defined by {f , g}, {h, k} = [f , h] + [g, k], {f , g}, {h, k} ∈ H 2 . T is called a closed
linear relation in H 2 if T is a (closed) linear manifold in H 2 . Such T is often considered
as a graph of (closed) linear (multivalued) operator. To any matrix-valued Nevanlinna
function, there is associated in a natural way a reproducing kernel Hilbert space. This
Hilbert space provides a model for a simple symmetric not necessarily densely defined
operator (see [5]).
We define the Cn
B to be the range R(B) in Cn×1 endowed with the inner product
[Bc, Bd] = d∗ Bc, where B is an entire function and c, d ∈ C. We note that Cn
B is equal to
the space L(B). As usual, L2 (dσ ) is the Hilbert space of all n×1 vector functions f de
fined on R such that f 2Σ = R f (t)∗ dΣ(t)f (t) < ∞ (see [5]). The theory of Hilbert space
of entire functions is a detailed description of eigenfunction expansions associated with
formally selfadjoint differential operators, then one can construct the expansion theorem for the reproducing kernels associated with the Hilbert space. In Section 2, we
collect some basic and essential observations (see [5, 6]), while in Section 3, we give the
main idea of de Branges space. In Section 4, we define the H(A, B) spaces when A and B
are entire functions. Finally, in Section 5, we give a description of the spectral functions
for the case we study.
2. Preliminaries. Let H be a Hilbert space, then L[H] is a set of bounded linear
operators from H to H. Furthermore, U is unitary if U −1 = U ∗ ⇒ U unitary operator,
D(U ) = R(U) = H, V is isometric if V −1 ⊂ V ∗ → V operator, S is symmetric if S ⊂ S ∗ ,
A is selfadjoint if A = A∗ , T is closed ρ(T ) = { ∈ C | (T − )−1 ∈ L(H)} resolvent set
(closed), σ (T ) = C \ ρ(T ) spectrum (open), and RT () = (T − )−1 , ∈ ρ(T ).
1704
A. A. EL-SABBAGH
When A = A∗ , put
H0 = H A(0),
A∞ = {0} × A(0) = {f , g} ∈ A | f = 0 ,
(2.1)
A 0 = A A∞ .
Then
A = A0 ⊕ A∞ ,
(2.2)
A0 is a selfadjoint operator (hence densely defined) in H0 :
H0
H0
A0
→
A(0)
A(0)
(2.3)
A∞
0 →
all of A(0).
Hence
−1
⊕ 0, where 0 is a zero operator on A(0)
(A − )−1 = A0 − −1
= A0 − P0 , P0 = orth. proj. of H onto H0 ,
ρ(A) = ρ A0 ⊃ C0 .
(2.4)
Let (Et0 ) be the orthogonal spectral family for A0 on H0 . Put
Et = Et0 ⊕ 0 = Et0 P0 ,
(2.5)
Et is called the orthogonal spectral family for A in H.
α β
For a 2 × 2 matrix M = γ δ , MT ≡ {{αf + βg, γf + δg} | {f , g} ∈ T }. The matrix M
is called a multiplier if α, β, γ, δ ∈ C. Then
0 1
1
0
1 0
T.
(2.6)
T −λ =
,
αT =
T,
T −1 =
1 0
−λ 1
0 α
Observe E∞ = P0 ,
RA () =
dEt0 P0
− RA0 ()P0 ,
t −
RA ()∗ = RA ,
dEt
=
t −
(2.7)
RA () − RA (λ) = ( − λ)RA ()RA (λ),
−RA () → P0 ,
→ ∞ along imaginary axis.
Now, consider S ⊂ S ∗ . Put M = M (S ∗ ) = {{f , g} ∈ S ∗ | g = f }. Note that D(M ) =
R(M ) = γ(S ∗ − ) = R(S − )⊥ . See [1, 2, 4, 25, 26, 27, 28, 29].
Now, the aim of this part is to associate to certain Hermitian operators a generalized
Fourier analysis. Let A and B be entire functions, Cn×n -valued such that the
KN (λ, u) =
A(λ)B ∗ (ω) + B(λ)A∗ (ω)
π i λ − ω∗
(2.8)
ON THE EXPANSION THEOREM DESCRIBED BY H(A, B) SPACES
1705
is positive such that, on R,
A(λ)B ∗ (λ) + B(λ)A∗ (λ) = 0,
A(0) = I,
B(0) = 0,
(2.9)
and let H(A, B) be the associated reproducing kernel Hilbert space of Cn×n -valued funcH
zf (z) is a closed symmetric operator in H(A, B) and, for any
tions. The operator z →
ω ≠ ω∗ ,
Dim(H − ωI)⊥ = n.
(2.10)
Moreover, if F is in H(A, B) and E(ω) = (A, B)(ω) is invertible, then
F (z) − E(z)E −1 (ω)F (ω)
z−ω
(2.11)
belongs to H(A, B).
Define
F (z) − E(z)E −1 (ω)F (ω)
,
z−ω
−1
R(ω)F = E(z)Rω E F .
R(ω)F =
(2.12)
R(ω) satisfies the resolvent equation. Moreover, R(ω) is bounded from H(A, B) into
H(A, B) and thus there is a relation T such that
(T − ωI)−1 = R(ω).
(2.13)
Then,
−1 F (z) − (A + B)(z)(A + B)−1 ω∗ F ω∗
∗
=
,
T − ωI
z − ω∗
−1
∗
−1
− i(T − ω)−1 − i ω∗ − ω T ∗ − ω∗
(T − ω)−1 ≥ 0.
i T −ω
(2.14)
Graph H = Graph T ∩ Graph T ∗ and ρ(T ) ⊂ C− .
In general, H(A, B) is not resolvent invariant and it is of interest to look for n × nvalued functions S(z) such that
F →
F (z) − S(z)S −1 (ω)F (ω)
.
z−ω
(2.15)
We denote by RS the above operator (2.15) (see [6, 7, 8]) which is a bounded operator
from H(A, B) into itself. We then call the space RS -invariant.
Associated to such a pair (A, B) will be the following structure problem.
Definition 2.1. Given a pair (A, B) of Cn×n -valued functions as above, the structure problem associated to (A, B) consists in finding all pairs (A , B ) with the same
properties and such that
(A, B) = (A , B )θ for some entire
0 I I 0
inner θ .
(2.16)
1706
A. A. EL-SABBAGH
This problem is a particular case of the inverse scattering problem defined in [6].
Note that H(A , B ) is contractively included in H(A, B) since
A(λ)B ∗ (ω) + B(λ)A∗ (ω)
A (λ)B (ω)∗ + B (λ)A∗ (ω)
=
iπ λ − ω∗
−iπ λ − ω∗
J − θ (λ)Jθ (ω)∗ A
+ A (λ), B (λ)
.
−iπ λ − ω∗
B
(2.17)
Theorem 2.2. Let (A, B) and H(A , B ) be as in (2.16) and suppose the space
H(A , B )RS is invariant. Then H(A, B) is RS -invariant.
Proof. The structure problem associated to a given space H(A, B) thus consists in
finding a family of invariant subspaces of H(A, B). We distinguish two cases.
Case 1. (A, B) is the upper part of a 0I 0I inner entire function. (This will happen if
and only if H(A, B) is resolvent invariant.)
Case 2. (A, B) is not a part of a 0I 0I inner entire function.
In the first case, using Potapov’s theorem, we associate to (A, B) a family (At , Bt )
which is not enough to define a Fourier analysis (see [7, 8, 12, 15, 16, 17, 18, 20, 21, 22,
24, 30]).
3. De Branges space. Let H be a Hermitian operator. We want to associate to H a
Fourier analysis, which really means that we want a model for H in a space of analytic
functions in terms of invariant subspaces (this last notion has to be precise).
The first step is to get a model for the operator H.
Theorem 3.1. Let H be a Hermitian operator, closed, simple, and with equal and
finite deficiency indices (n, n). Suppose that the graph of H is the intersection of the
graph of T and T ∗ , where T is a relation, extending H, with spectrum in the open lower
half-plane and such that, for ω in C+ ,
−1/2
−1
i T ∗ − ω∗
− i(T − ω)−1 − i ω − ω∗ T ∗ − ω∗
(T − ω)−1 ≥ 0.
(3.1)
Proof. There exist n × n entire functions A and B such that (A − B) is invertible in
C+ , (A + B) is invertible in C− such that (A − B)−1 (A + B) is inner, and H is unitarily
equivalent to multiplication by λ in H(A, B), the reproducing kernel Hilbert space with
reproducing kernel
A(λ)B ∗ (ω) + B(λ)A∗ (ω)
.
iπ λ − ω∗
(3.2)
An interesting feature of the proof of the theorem is that
(A − B)−1 (A + B)(λ) = I + 2π iλJ(λ)J(0)∗ ,
(3.3)
−1
J(λ) = Jω I + (α − ω)(T − α)
(3.4)
where
and Jω is a first part of the operator in (3.1).
ON THE EXPANSION THEOREM DESCRIBED BY H(A, B) SPACES
1707
The space H(A, B) is not in general resolvent invariant, and it is of interest, in order
to define invariant subspaces, to characterize functions S, entire and n×n-valued, such
that H(A, B) is closed under the operator RS (u),
f (λ) − S(λ)S −1 (ω)f (ω)
.
RS (u)f =
λ−ω
(3.5)
Such conditions are stated in [6, 8].
To define the Fourier analysis associated to H, we first define the following structure
problem for A, B.
Definition 3.2. Let A, B be entire n × n-valued functions as above. The structure
problem associated to H(A, B) consists in finding all pairs (A , B ) such that an H(A , B )
space exists and such that (A, B) = (A , B )θ , for some 0I 0I inner entire function θ .
Note that (3.3) implies that H(A , B ) is contractively included in H(A, B).
The space H(A, B) will be RS -invariant as soon as H(A , B ) is RS -invariant.
The question at hand can thus be described as follow. Given a medium of the operator H when enough RS -invariant subspaces are known. The answer to this question is
different according to if S can be chosen to be I, there or not, see [3, 6, 13, 14, 17, 18, 19].
B
When S is equal to I, there exist matrix-valued functions C and D such that M = A
C D
0 I is I 0 inner and entire.
Theorem 3.3. Let M(λ) be a J-inner entire function. Then
t
M(λ) =
exp iλH(u)du,
(3.6)
0
where
H(u)J ≥ 0,
TT H(u)J = 1;
(3.7)
and let
t
M(t, λ) =
∂M
= iλM · H(u).
∂t
exp iλH(u)du,
0
(3.8)
Then (A(t, λ), B(t, λ)) is a solution of the structure problem associated to A, B.
Proof. When J = 0I 0I ,
J − M(t, λ)JM(t, ω)
=
iπ λ − ω∗ 2π
t
M(u, λ)
0
H(u)J ∗
M (u, ω)du
2π
(3.9)
and so the map
f →
M(u, λ)
0
H(u)J
f (u)du
2π
is a partial isometry from L2 (H, [0, )) into H(M).
(3.10)
1708
A. A. EL-SABBAGH
Let
F (z) =
M(u, λ)
0
H(u)J
f (u)du.
2π
(3.11)
R0 F belongs to H(M) and thus there exists an element g in L2 (H, [0, ]) such that
R0 F (z) =
M(u, λ)
0
H(u)J
g(u)du
2π
(3.12)
(see [5, 9, 10, 11, 15, 20]).
The functions f and g are linked by (g is chosen in the orthogonal of the kernel of
the partial isometry)
1
1
c
H(u)Jg(u)du =
c
H(c) − H(u) H(u)Jf (u)du.
(3.13)
When S cannot be chosen to I (and thus the space H(A, B) is not resolvent invariant),
the situation is more involved. One cannot construct from (A, B) the mass function
H(u)J and has to state as a hypothesis the enough solutions to the structure problem
for A, B, that is, the existence of a family H(A(t), B(t)), t > 0, such that for every t > 0,
H(A(t), B(t)) is contractively included in H(A, B). One has also to state as a hypothesis
the existence of a mass function H(u) such that a weakened version of (3.9) holds,
namely, for any a, b > 0, z, ω in C,
A(b, z), B(b, z) J A(b, z), B(b, z)∗ − A(a), B(a) JA (a, ω), B(a, ω)∗
(3.14)
b
= −i z − ω∗
A(t, z), B(t, z) H(u)J A(t, ω), B(t, ω) dt.
a
Theorem 3.4. Under such hypothesis, the map
b
f →
A(t, z), B(t, z) H(u)du
(3.15)
a
is a partial isometry from L2 (HJ, [a, b]) onto the completion for H(A(b), B(b)) of
H(A(a), B(a)).
When a may be chosen to be zero (i.e., when some limit process is justified), a partial
isometry from L2 (HJ, [a, b]) onto H(A, B) = H(A(b), B(b)) is obtained.
Proof. Let
b
F (z) =
A(t, z), B(t, z) H(t)Jf (t)du
(3.16)
a
be in the orthogonal complement of the partial isometry and suppose F (0) = 0. Then
F (z)/z belongs to H(A, B) and thus there exists a g such that
b
θ(z) =
A(t, z), B(t, z) H(t)Jf (t)du,
0
f and g are linked by (3.13) (see [6, 23, 30]).
(3.17)
ON THE EXPANSION THEOREM DESCRIBED BY H(A, B) SPACES
1709
4. Theory of H(A, B) spaces. In order to prove the expansion theorem described in
Section 3, a more detailed analysis of H(A, B) spaces is needed (see [3, 5, 7, 8]).
We first recall that a function analytic in the upper half-plane C+ is said of bounded
type if it is a quotient of two bounded analytic functions in C+ . It can be then written
as
dµ(t) tz + 1
f (z) = eizh B(z) · exp i
,
t2 + 1 z − t
(4.1)
where B is Blaschke product h ∈ R, and (|dµ|/(t 2 + 1)) < ∞.
Definition 4.1. The number h is called the mean type of f . When h is negative,
then it is said to be of nonpositive mean type. The number h is negative if and only if,
for any ε > 0,
lim e−εy f (iy) = 0.
(4.2)
y→+∞
The interest in the functions of bounded type with nonpositive mean type is that the
Cauchy formula holds for such functions, provided some regularity is satisfied on the
real line.
Theorem 4.2. Let f be analytic in C+ of bounded type and nonpositive mean type in
C and suppose f has a continuous extension to the closure of C+ . Then
+
|f |2 (t)dt < ∞.
(4.3)
Proof. We have
2π if (z) =
∞
−∞
f (t)
dt = 0
t −z
for z in C+
(4.4)
(see [1, 2, 3, 4, 6, 7, 8]).
5. Spectral function. We defined the Nevanlinna class Nn as the class of all n × n
matrix functions N(), which are holomorphic in C0 , satisfying N()∗ = N(), ∈ C0 ,
and for which the kernel
KN (, λ) =
N() − N(λ)∗
−λ
,
, λ ∈ C0 , ≠ λ,
(5.1)
is nonnegative. The reproducing kernel Hilbert space associated with the kernel KN (, λ)
(see [7, 8]) is denoted by L(N). For general information concerning reproducing kernel
Hilbert spaces, we refer to Aronszajn [3]. It is well known that for each N() ∈ Nn , there
exist n×n matrices A and B with A = A∗ , B = B∗ ≥ 0, and a nondecreasing n×n matrix
function Σ on R with R (t 2 + 1)−1 dΣ(t) < ∞ such that
N() = A + B +
R
t
1
− 2
dΣ(t),
t − t +
∈ C0 .
(5.2)
1710
A. A. EL-SABBAGH
With this so-called Riesz-Hergoltz representation, the kernel KN (, λ) takes the form
KN (, λ) = B +
R
dΣ(t)
,
(t − ) t − λ
, λ ∈ C0
(5.3)
to be the range R(B) of B in
(see [10, 13, 29, 30, 31, 32, 33]). We define the space Cn
B̃
is equal to the
Cn×1 endowed with the inner product [Bc, Bd] = d∗ Bc. We note that Cn
B̃
space L(B). As usual, L2 (dΣ) is the Hilbert space of all n×1 vector functions f defined
on R such that
f (t)∗ dΣ(t)f (t) < ∞.
(5.4)
f 2Σ =
R
Recall that by the Stieltjes-Liversion formula, the functions (t − )−1 c, c ∈ Cn×1 and
∈ C0 , are dense in L2 (dΣ). The scalar version of the next result can be found in [8].
Proposition 5.1. Let () ∈ Nn have the integral representation (5.2). The Hilbert
space L(n) is isomorphic to the space of all n × 1 vector functions F () of the form
F () = Bc +
R
dΣ(t)f (t)
,
t −
∈ C0 ,
(5.5)
where c ∈ Cn×1 , f ∈ L2 (dΣ) are uniquely determined by F (), with norm given by
F 2 = c ∗ Bc + f 2Σ .
(5.6)
Proof. Let F () be an element of the form (5.5). We first check that Bc and f are
uniquely determined by F (). Indeed, if F () admits two such representations with
f ∈ L2 (dΣ), c ∈ Cn×1 and f˜ ∈ L2 (dΣ), c̃ ∈ Cn×1 , we have
B(c̃ − c) =
R
−1 dΣ(t) f (t) − f˜(t)
= f (t) − f˜(t), t − .
Σ
t −
(5.7)
Letting → ∞, we obtain Bc = Bc̃ and f = f˜. The set of functions of the form (5.5) with
norm (5.6) is easily seen to be a Hilbert space, which we denote by K. The representation
(5.3) shows that the function → KN (, λ)d belongs to K for any d ∈ Cn×1 and λ ∈ C0 .
Moreover, for F () of the form (5.5), we have the reproducing property of the kernel
KN (, λ),
F , KN (·, λ)d = d∗ F (λ),
d ∈ Cn×1 ,
(5.8)
where [·, ·] denotes the inner product associated with the norm (5.6). The uniqueness
of the reproducing kernel Hilbert space with reproducing kernel KN (, λ) implies that
K and L(N) are isomorphic.
⊕ L(N − B). The mapping
Note that we may write L(N) = L(B) ⊕ L(N − B) = Cn
B̃
defined by
f →
R
dΣ(t)f (t)
,
t −
f ∈ L2 (dΣ),
(5.9)
ON THE EXPANSION THEOREM DESCRIBED BY H(A, B) SPACES
1711
is an isometry from L2 (dΣ) onto L(N −B) and the mapping defined by (5.5) is an isom⊕ L(N − B) onto L(N). The elements of L(N) are n × 1 vector functions,
etry from Cn
B̃
which are defined and holomorphic on C0 . This follows from the fact that L(N) is a
reproducing kernel Hilbert space, and also from (5.5). We will identify L(N) with the
⊕ L2 (dΣ). For any n × 1 vector function F (), holomorphic on C0 , and any
space Cn
B̃
λ ∈ C0 , we define the operator Rλ by
F () − F (λ)
,
Rλ F () =
−λ
∈ C0 , ≠ λ,
Rλ F (λ) = F (λ).
(5.10)
For Hilbert spaces K, R, we denote by L(K, R) the space of all bounded linear operators
from K to R; we write L(K) = L(K, K).
Proposition 5.2. The following items are satisfying:
(i) for all λ ∈ C0 , Rλ ∈ L(L(N)). In fact, Rλ F ≤ | Im λ|−1 F , F ∈ L(N);
(ii) for all λ ∈ C0 , (Rλ )∗ = Rλ ;
(iii) the resolvent identity Rλ − Rµ = (λ − µ)Rµ Rλ holds for all λ, µ ∈ C0 .
Proof. Let F () ∈ L(N) have representation (5.5). Then for λ ∈ C0 ,
Rλ F () =
R
dΣ(t)f (t)
,
(t − )(t − λ)
(5.11)
which on account of Proposition 5.1 implies that Rλ F ∈ L(N), λ ∈ C0 . By (5.11),
2 f (t)
f (t) ∗
Rλ F 2 = f (t) =
dΣ(t)
t −λ
t −λ
t −λ
Σ
R
≤ | Im λ|−2 f (t)∗ dΣ(t)f (t) = | Im λ|−2 f 2Σ ≤ | Im λ|−2 F 2 ,
(5.12)
R
for λ ∈ C0 , which shows that Rλ is a bounded operator in L(N); this proves (i). Items
(ii) and (iii) also follow from (5.11).
Combining (ii) and (iii) of Proposition 5.2, we obtain the identity
Rλ − Rµ∗ = λ − µ Rµ∗ ,
λ, µ ∈ C0 .
(5.13)
Conversely, if Rλ ∈ L(L(N)) satisfies (5.13), then the inequality in (i), (ii), and (iii) follow.
For a selfadjoint relation (i.e., multivalued operator) A in a Hilbert space K, A(0) =
{ϕ ∈ K | {0, ϕ} ∈ A} stands for the multivalued part of A and As = A ∩ (K A(0))2
is the operator part of A, which is a (densely defined) selfadjoint operator in K A(0)
with D(As ) = D(A). In the following theorem, we assume that N() ∈ Nn and that it
has the integral representation (5.2).
Theorem 5.3. The operator Rλ , λ ∈ C0 , is the resolvent operator of the selfadjoint
relation A in L(N) given by
A = {F , G} ∈ L(N)2 | G() − F () = c, ∀ ∈ C0 , for some c ∈ Cn×1 .
(5.14)
1712
A. A. EL-SABBAGH
The multivalued part of A is equal to A(0) = L(N) ∩ Cn×1 = R(B). Representing F () ∈
L(N) by (5.5),
(i)
dΣ(t)f (t)
for some f ∈ L2 (dΣ) with tf (t) ∈ L2 (dΣ),
F ∈ D(A) ⇐⇒ F () =
t −
R
(5.15)
(ii)
As F () = F () +
R
dΣ(t)f (t) =
R
dΣ(t)tf (t)
,
t −
F ∈ D(A).
(5.16)
Proof. Proposition 5.2 implies that B = {{Rµ H, (I +µRµ )H} | H ∈ L(N)} is a selfadjoint relation in L(N) which is independent of µ ∈ C0 . We have Rλ = (B − λ)−1 , that is,
Rλ is the resolvent of B. It is easy to see that B is contained in the right-hand side of
(5.14). To prove the inclusion A ⊂ B, take {F , G} ∈ L(N)2 for which G()−F () = c for
some c ∈ Cn×1 and put H() = G() − µF (). Then H() ∈ L(N), (Rµ H)() = F (), and
((I + µRµ )H)() = G() and hence {F , G} ∈ B. This proves B = A and the first part of
the theorem. The item about the multivalued part of A is clear. Let F () ∈ D(A) have
the representation (5.5). Then Bc = 0 since D(A) is orthogonal to A(0). By (5.14), there
exists a constant d ∈ Cn×1 so that F () + d ∈ L(N) and hence for some e ∈ Cn×1 and
g ∈ L2 (dΣ),
R
dΣ(t)f (t)
+ d = Be +
t −
R
dΣ(t)g(t)
.
t −
Applying Rα with α ∈ C0 to both sides, we obtain
dΣ(t) g(t) − tf (t)
=0
(t − )(t − α)
R
(5.17)
(5.18)
which implies g(t) = tf (t) and hence tf (t) ∈ L2 (dΣ). Therefore, F has the representa
tion in (i), the integral R dΣ(t)f (t) exists, and
F () =
R
dΣ(t)f (t)
=−
t −
R
dΣ(t)f (t) +
R
dΣ(t)tf (t)
.
t −
(5.19)
This shows that F () + R dΣ(t)f (t) ∈ L(N) and is orthogonal to A(0) so that (ii) has
been proved. As to the converse of (i), let F () ∈ L(N) have the indicated representation
for some f ∈ L2 (dΣ) with tf (t) ∈ L2 (dΣ). Then R dΣ(t)f (t) exists and
F () +
R
dΣ(t)f (t) =
R
dΣ(t)tf (t)
,
t −
(5.20)
where the right-hand side belongs to L(N). Therefore, by (5.14), F () ∈ D(A) (see [6, 9]).
Several observations and proofs in this section are due to A. Dijksma, H. de Snoo, and
P. Bruinsma.
Acknowledgment. I would like to thank P. Bruinsma, H. de Snoo, and A. Dijksma
for their support and advice during my stay in Groningen University, the Netherlands.
ON THE EXPANSION THEOREM DESCRIBED BY H(A, B) SPACES
1713
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A. A. El-Sabbagh: Department of Mathematics, Faculty of Engineering, Zagazig University, 108
Shoubra Street, Shoubra, Cairo, Egypt
E-mail address: alysab1@hotmail.com